Rational version of Archimedes symplectomorphysm and birational Darboux coordinates on coadjoint orbit of $GL(N,C)$

Abstract

A set of all linear transformations with a fixed Jordan structure J is a symplectic manifold isomorphic to the coadjoint orbit O(J) of GL(N, C).Any linear transformation may be projected along its eigenspace to (at least one) coordinate subspace of the complement dimension. The Jordan structure J of the image is defined by the Jordan structure J of the pre-image, consequently the projection O(J) → O( J ) is the mapping of the symplectic manifolds.It is proved that the fiber E of the projection is a linear symplectic space and the map O(J)The iteration of the procedure gives the isomorphism between O(J) and the linear symplectic space, which is the direct product of all the fibers of the projections. The Darboux coordinates on O(J) are pull-backs of the canonical coordinates on the linear spaces in question.